International Mathematical Olympiad · 2021

Problems

Problem 1Let $n \ge 100$ be an integer. Ivan writes the numbers $n, n+1, \ldots, 2n$ each on a different card. He then shuffles these $n+1$ cards and divides them into two piles. Prove that at least one of the piles contains two cards whose numbers sum to a perfect square.Solutions: 1
Problem 2Show that the inequality $\sum_{i=1}^n\sum_{j=1}^n\sqrt{|x_i-x_j|}\le\sum_{i=1}^n\sum_{j=1}^n\sqrt{|x_i+x_j|}$ holds for all real numbers $x_1,x_2,\ldots,x_n$ .Solutions: 1
Problem 3Let $D$ be an interior point of the acute triangle $ABC$ with $AB>AC$ so that $\angle DAB=\angle CAD$ . The point $E$ on the segment $AC$ satisfies $\angle ADE=\angle BCD$ , the point $F$ on the segment $AB$ satisfies $\angle FDA=\angle DBC$ , and the point $X$ on the line $AC$ satisfies $CX=BX$ . Let $O_1$ and $O_2$ be the circumcenters of the triangles $ADC$ and $EXD$ , respectively. Prove that the lines $BC$ , $EF$ , and $O_1O_2$ are concurrent.Solutions: 1
Problem 4Let $\Gamma$ be a circle with centre $I$ , and $ABCD$ a convex quadrilateral such that each of the segments $AB$ , $BC$ , $CD$ and $DA$ is tangent to $\Gamma$ . Let $\Omega$ be the circumcircle of the triangle $AIC$ . The extension of $BA$ beyond $A$ meets $\Omega$ at $X$ , and the extension of $BC$ beyond $C$ meets $\Omega$ at $Z$ . The extensions of $AD$ and $CD$ beyond $D$ meet $\Omega$ at $Y$ and $T$ , respectively. Prove that $AD+DT+TX+XA=CD+DY+YZ+ZC.$ Solutions: 1
Problem 5Two squirrels, Bushy and Jumpy, have collected $2021$ walnuts for the winter. Jumpy numbers the walnuts from $1$ through $2021$ and digs $2021$ little holes in a circular pattern in the ground around their favourite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid no attention to the numbering. Unhappy, Jumpy decides to reorder the walnuts by performing a sequence of $2021$ moves. In the $k$ -th move, Jumpy swaps the positions of the two walnuts adjacent to walnut $k$ . Prove that there exists a value of $k$ such that, on the $k$ -th move, Jumpy swaps some walnuts $a$ and $b$ with $a<k<b$ .Solutions: 1
Problem 6Let $m\ge2$ be an integer, $A$ a finite set of (not necessarily positive) integers, and $B_1,B_2,B_3,\ldots,B_m$ subsets of $A$ . Suppose that for every $k=1,2,\ldots,m$ the sum of the elements of $B_k$ is $m^k$ . Prove that $A$ contains at least $m/2$ elements.Solutions: 1